It seems to me that the solution provided here is wrong, because I think the solution satisfies both formulas. Would you please prove me wrong if I am mistaken? This does not mean that I am claiming that the formulas are equivalent. So please provide another counter example if the one here is wrong, and if it is correct please explain why.

1 Antwort

The meaning of these formulas should be clear: In general, AG a holds in all states where all outgoing infinite paths always satisfy a (which is only the case in s2), and AF phi holds in all states where all outgoing infinite paths at least once satisfy phi. In case of phi=(AG a), we seek the states where all outgoing infinite paths at least once visit state s2. That is true in states s1 and s2, but not in state s0, since there is the path that always loops in s0.

The formula AGFGa is equivalent to AFG a (you can use the LTL prover on http://es.cs.uni-kl.de/tools/teaching/TemporalLogicProver.html to prove the equivalence G F G a <-> F G a). AFGa holds in all states where all outgoing infinite paths finally always satisfy a. That is true in state s0, since the first path which always loops in s0 satisfies the property, and the second path that leaves s0 (after looping there for some finite time), will have to reach s2 where it loops forever and therefore, it will also finally always satisfy a.

Hence, AGFGa holds in state s0, but AFAG a does not, and therefore the structure distinguishes between the two formulas.

Kommentare

Thanks! The explanation is persuasive, but then another question rises. Finding a solution for this question does not seem to be intuitive, at least for me. It makes sense to analyze the answer but to come up with an answer seems hard, Do you have any tips on that?

Well, there is an algorithmic way to decide these problems, but I would say that this would be overkill. It is best to carefully read the formulas and to try to intuitively find a difference. If that does not help, split it into a present and future part (by unwinding temporal operators).